Ana Proceedings of the American Mathematical Society A note on the bilinear Littlewood-Paley square function

A note on the bilinear Littlewood-Paley square function

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138
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Proceedings of the American Mathematical Society
DOI:
10.1090/S0002-9939-10-10233-0
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```PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 138, Number 6, June 2010, Pages 2095–2098
S 0002-9939(10)10233-0
Article electronically published on January 29, 2010

A NOTE ON THE BILINEAR
LITTLEWOOD-PALEY SQUARE FUNCTION
PARASAR MOHANTY AND SAURABH SHRIVASTAVA
(Communicated by Michael T. Lacey)

Abstract. In this paper, we give an elementary proof of boundedness of the
smooth bilinear Littlewood-Paley square function.

1. Introduction
The study of bilinear multiplier operators has recently received a great deal of
attention with the work of Lacey and Thiele [4], [5]. They proved the boundedness
of the bilinear Hilbert transform and established the answer to a long-standing
conjecture of A.P. Calderón.
One of the important themes in the study of linear multipliers is Rubio de Francia’s result on Littlewood-Paley square functions [7]. His result is:

1
(
|SIj f |2 ) 2 p ≤ Cf p
j

for p ≥ 2, where SIj f = (χIj fˆ)∨ and the Ij ’s are disjoint intervals. First, Lacey
[2] addressed this problem in the bilinear setting. Later, Bernicot [1] proved a
version of Rubio de Francia’s result in the bilinear case. The proofs of these two
results are very intricate. In [2] Lacey posed a problem regarding the validity of
the boundedness of smooth square functions for certain values of p. Surprisingly,
the answer to his question can be given even without any use of time-frequency
analysis. The purpose of this note is to give an answer to Lacey’s question.
Let K be a smooth bump function deﬁned on Rd such that K̂ is supported in the
unit cube of Rd . For n ∈ Zd , let Kn be the function deﬁned by Kˆn (ξ) = K̂(ξ − n).
For f, g ∈ S(Rd ) consider the bilinear operator

f (x − y)g(x + y)Kn (y)dy.
Sn (f, g)(x) =
Rd

Let S(f, g) denote the bilinear Littlewood-Paley square function associated with
this sequence of operators, i.e.

1
S(f, g)(x) = (
(1.1)
|Sn (f, g)(x)|2 ) 2 .
n

In [2] Lacey proved the following result:
Received by the editors September 25, 2009.
2000 Mathematics Subject Classiﬁcation. Primary 42A45,;  42B15, 42B25.
Key words and phrases. Bilinear multiplier operators, Littlewood-Paley square function.
c
2010
American Mathematical Society
Reverts to public domain 28 years from publication

2095

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2096

PARASAR MOHANTY AND SAURABH SHRIVASTAVA

Theorem 1.1 ([2]). For all 2 ≤ p1 , p2 ≤ ∞ with

1
p1

+

1
p2

= 12 ,

S(f, g)2 ≤ cf p1 gp2 ,

(1.2)

where the constant C depends on the function K.
The problem for nonsmooth symbols remained open for a long time. Recently
Bernicot [1] addressed this problem and obtained the boundedness of the nonsmooth
Littlewood-Paley square function associated with a sequence of intervals satisfying
certain conditions. In particular, the bilinear analogue of Carleson’s square function
has been obtained.
In the proof of Theorem 1.1 the exponent 2, the decay of the bump function K,
and the support condition on K̂ play important roles. In [2] the author posed a
question about the boundedness of S(f, g) into Lp3 (Rd ) for exponents p3 > 2. The
answer follows from an adaptation of the techniques used by Rubio de Francia in
[6]. Moreover, with this technique one gets the result for a much wider range of the
exponent p3 and with weaker restrictions on K. Also, we will show that p1 , p2 ≥ 2
is a necessary condition. But still we do not have any idea about the sharpness of
the bound for the exponent p3 .

2. Smooth bilinear Littlewood-Paley square function
In this section we obtain the result of Theorem 1.1 for a larger range of exponents
p1 , p2 , and p3 using the ideas from [6].
Theorem 2.1. Let m ∈ S(R). For k ∈ Z deﬁne mk (ξ) = m(ξ − k) and let Sk be
the bilinear multiplier operator associated with mk . Then for 2 < p1 , p2 ≤ ∞ and
4/3 < p3 ≤ ∞ satisfying p11 + p12 = p13 , we have
S(f, g)p3 ≤ Cf p1 gp2 .

(2.1)

Proof. In order to prove inequality (2.1), it is enough to prove that for almost every
x ∈ R, S(f, g)(x) satisﬁes the pointwise estimate
(2.2)


1
S(f, g)(x) ≤ C M (|f |2 , |g|2 )(x) 2 = CM2 (f, g)(x)

where M is the bilinear Hardy-Littlewood maximal operator, given by

1
M (f, g)(x) = sup
|f (x − y)g(x + y)|dy,
x∈I |I| I
and C is a constant independent of f and g. The above estimate gives us the desired result upon using the boundedness of the bilinear Hardy-Littlewood maximal
operator M proved by Lacey in [3], which states that M (f, g)p3 ≤ Cf p1 gp2
where p13 = p11 + p12 and 2/3 < p3 ≤ ∞, 1 < p1 , p2 ≤ ∞.
Let a = {ak } be a ﬁnite sequence in l2 (Z) with norm one. Then we prove that
for a.e. x ∈ R,

ak Sk (f, g)(x)| ≤ CM2 (f, g)(x),
|
k

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THE BILINEAR LITTLEWOOD-PALEY SQUARE FUNCTION

2097

which gives the claimed estimate (2.2) for the square function S(f, g). Consider

 
ak Sk (f, g)(x) =
ak
f (x − y)g(x + y)m̌k (y)dy
k

R

k
=
R

f (x − y)g(x + y)m̌(y)

R

ak e2πiky dy

k


=



f (x − y)g(x + y)m̌(y)h(y)dy,

where h is the Fourier transform of the sequence a and is a periodic function with
unit L2 (I) norm, for I = [−1/2, 1/2]. Hence


ak Sk (f, g)(x) =
f (x − y)g(x + y)m̌(y)h(y)dy
R

k


=

+
I

∞ 

n=1


f (x − y)g(x + y)m̌(y)h(y)dy.

2n I\2n−1 I

CN
Since m ∈ S(R), there exists a constant CN such that |m̌(y)| ≤ (1+|y|)
N for all
N ∈ N. Using this decay estimate for m̌ and Hölder’s inequality, we obtain

∞
1/2




ak Sk (f, g)(x) ≤ CN
2−nN 2n/2
|f (x − y)g(x + y)|2 dy
k

≤ CN

n=0
∞


2n I


1
2−n(N −1) M (|f |2 , |g|2 )(x) 2

n=0

≤ CN M2 (f, g)(x).
This completes the proof in one dimension.



Note that this result covers a wider range of exponents than mentioned by Lacey
in dimension 1. If the dimension d > 1, then using the same idea we obtain that
the above theorem holds when p3 ≥ 2 and 2 < p1 , p2 ≤ ∞, as the d-dimensional
bilinear Hardy-Littlewood maximal operator M2 is easily seen to be bounded for the
above-mentioned range as a consequence of Hölder’s inequality and interpolation.
So, this completely answers the question posed by Lacey.
We would like to make some remarks:
C
(1) In the proof of Theorem 2.1 we only need a (1+|y|)
2 type decay for the
function m̌, and no support condition on m is required.
(2) For the case where p3 = 2 and either of p1 , p2 is 2, this technique will yield
that the operator S is of weak type as the operator M2 is of weak type
for these exponents. However, Lacey’s result, Theorem 1.1, provides better
estimates for these end-point cases when m has compact support.

Now we shall show that p1 , p2 ≥ 2 is a necessary condition in Theorem 2.1. This
can be achieved using a well known example adapted to our setting.
Proposition 2.2. If Theorem 2.1 holds true for some indices p1 , p2 , p3 , then it is
a necessary condition that p1 , p2 ≥ 2.

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2098

PARASAR MOHANTY AND SAURABH SHRIVASTAVA

Proof. Let fˆN = χ[0,N ] , ĝ = χ[0,1/2] and m ∈ S(R) be such that supp(m) ⊆ [0, 1].
Observe that
 
Sk (fN , g)(x) =
χ[0,N ] (ξ − η)χ[0,1/2] (η)m(ξ − 2η − k)dηe2πiξx dξ.
R

R

An easy computation will show that for 1 ≤ k ≤ N − 2 and η ∈ [0, 1/2], ξ − 2η − k ∈
[0, 1] implies that ξ − k ∈ [0, 2] and ξ − η ∈ [0, N ]. Hence, for 1 ≤ k ≤ N − 2, we
have


χ[0,N ] (ξ − η)χ[0,1/2] (η)m(ξ − 2η − k)dη
Sk (fN , g)(ξ) =
R
1/2



χ[0,N ] (ξ − η)m(ξ − 2η − k)dη

=
0

=

1
χ[0,2] (ξ − k)
2



1

m(ξ − η − k)dη
0

= F̂ (ξ − k),
where F̂ (ξ) =

1
1
2 χ[0,2] (ξ) 0

m(ξ − η)dη. Thus for 1 ≤ k ≤ N − 2, we have
|Sk (fN , g)(x)| = |F (x)|.

If Theorem 2.1 holds true, then we have that
N 1/2 ≈ (

N
−2


|Sk (fN , g)(x)|2 )1/2 p3 ≤ S(fN , g)p3 ≤ CfN p1 gp2 .

k=1

This implies that


N 1/2 ≤ C  N 1/p1
where the constant C  does not depend on N . Hence we get that p1 ≥ 2. Similar

arguments can be given to show that p2 also satisﬁes the same condition.
References
[1] Bernicot, F.,
estimates for non smooth bilinear Littlewood-Paley square functions on R,
arXiv:0811.2854, to appear in Math. Ann.
[2] Lacey, M., On bilinear Littlewood-Paley square functions, Publ. Mat. 40 (1996), no. 2, 387–
396. MR1425626 (98c:42017)
[3] Lacey, M., The bilinear maximal functions map into Lp for 2/3 < p ≤ 1, Ann. of Math. (2)
151 (2000), no. 1, 35–57. MR1745019 (2001b:42015)
[4] Lacey, M., Thiele, C., Lp estimates on the bilinear Hilbert transform for 2 < p < ∞, Ann.
of Math. (2) 146 (1997), no. 3, 693–724. MR1491450 (99b:42014)
[5] Lacey, M., Thiele, C., On Calderón’s conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475–496.
MR1689336 (2000d:42003)
[6] Rubio de Francia, J. L., Estimates for some square functions of Littlewood-Paley type, Publ.
Sec. Mat. Univ. Autònoma Barcelona 27 (1983), no. 2, 81–108. MR765844 (86d:42018)
[7] Rubio de Francia, J. L., A Littlewood-Paley inequality for arbitrary intervals, Rev. Mat.
Iberoamericana 1 (1985), no. 2, 1–14. MR850681 (87j:42057)
Lp

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur,
Kanpur-208016, India
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur,
Kanpur-208016, India

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